Abstract
An exact formula for the various measure dimensions of attractors associated with contracting similitudes is given. An example is constructed showing that for more general affine maps the various measure dimensions are not always equal.
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M. F. Barnsley (1986):Fractal functions and interpolation. Constr. Approx.,2:303–329.
M. F. Barnsley, S. Demko (1985):Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London Ser. A,399:243–275.
M. F.Barnsley, S.Demko, J.Elton, J. S.Geronimo (to appear):Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. H. Poincaré.
T.Bedford (1984): Crinkly Curves, Markov Partitions and Dimensions. Ph.D. Thesis, University of Warwick.
M. F. Barnsley, V. Ervin, D. Hardin, J. Lancaster (1986):Solution of an inverse problem for fractals and other sets. Proc. Nat. Acad. Sci. U.S.A.,83:1975–1977.
S. Demko, L. Hodges, B. Naylor (1985):Construction of fractal objects with iterated function systems. Comput. Graphics,19:271–278.
P. Diaconis, M. Shahshahani (1986):Products of random matrices and computer image generation. Contemp. Math.,50:173–182.
K. J. Falconer (1985): The Geometry of Fractal Sets. Cambridge: Cambridge University Press.
J. D. Farmer, E. Ott, J. A. Yorke (1983):The dimension of chaotic attractors. Physica,70:153–180.
J. Hutchinson (1987):Fractals and self-similarity. Indiana Univ. Math. J.,30:731–747.
F. Ledrappier (1981):Some relations between dimension and Lyapunov exponents. Comm. Math. Phys.,81:229–238.
B. Mandelbrot (1982): The Fractal Geometry of Nature. San Francisco: Freeman.
J. M. Marstrand (1954):Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3),4:257–302.
P.Massopust (1986): Space Curves Generated by Iterated Function Systems. Thesis, Georgia Institute of Technology.
C. McMullen (1984):The Hausdorff dimension of general Sierpinski carpets. Nazoyo Math. J.,96:1–10.
S. Pelikan (1984):Invariant densities for random maps of the interval. Trans. Amer. Math. Soc.,281:813–815.
L. S. Young (1982):Dimension, entropy, and Lyapunov exponents. Ergodic Theory Dynamical Systems,2:109–124.
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Communicated by Michael F. Barnsley.
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Geronimo, J.S., Hardin, D.P. An exact formula for the measure dimensions associated with a class of piecewise linear maps. Constr. Approx 5, 89–98 (1989). https://doi.org/10.1007/BF01889600
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DOI: https://doi.org/10.1007/BF01889600